Finite Group Databasebeta

← C23×#8507 #19156133

Subgroups — C23×#8507 #19156133

Order 5888.

736 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#563C211yesmaximal, characteristic
2C23×#1992C212yesmaximal, complemented
3C23×#1959C211yesmaximal, characteristic
4C2×C23×#330C212yesmaximal, complemented
5C2×C23×#329C211yesmaximal, characteristic
6C23×#563—24no—
7C23×#563—22no—
8C23×#563C414yescomplemented
9C2×C23×C4⋊Q8—21no—
10C2×C23×#126—22no—
11C2×C23×#125C2212yes—
12C2×C23×#124C2211yescharacteristic
13C2×C23×#123—21no—
14C2×C23×C22⋊Q8—22no—
15C2×C23×#103—21no—
16C2×C23×#102—21no—
17C2×C23×#97C2211yescharacteristic
18C2×C23×#97C2212yes—
19C23×#384C2211yescharacteristic
20C23×#340—22no—
21C23×#340—22no—
22C23×#330—24no—
23C23×#329—22no—
24C22×Q8×C23—21no—
25C22×Q8×C23D411yescharacteristic
26C22×D4×C23D411yescharacteristic
27C23×C4×C23—21no—
28C22×C8×C23—22no—
29C22×C8×C23—21no—
30C23×C4⋊Q8—24no—
31C23×C4⋊Q8—24no—
32C23×#126—28no—
33C23×#126—28no—
34C23×#125—22no—
35C23×#125—22no—
36C23×#125C2×C414yes—
37C23×#125D414yes—
38C23×#124—22no—
39C23×#124C2×C412yes—
40C23×#124D412yes—
41C23×#123—24no—
42C23×#123—24no—
43C23×C22⋊Q8—28no—
44C23×C22⋊Q8—28no—
45C2×C4⋊C4×C23—22no—
46C2×C4⋊C4×C23—22no—
47C2×C4⋊C4×C23—22no—
48C2×C4⋊C4×C23—22no—
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23D412yes—
51C2×C4⋊C4×C23D412yes—
52C2×C22⋊C4×C23—22no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—22no—
55C2×C42×C23—21no—
56C2×C42×C23C2311yescharacteristic
57C23×#103—42no—
58C23×#102—42no—
59C23×#97—22no—
60C23×#97—22no—
61C23×#97—22no—
62C24×C23—21no—
63C2×Q8×C23—24no—
64C2×Q8×C23—21no—
65C2×Q8×C23—41no—
66C2×Q8×C23—41no—
67C2×Q8×C23—81no—
68C2×Q8×C23—81no—
69C2×Q8×C23C22⋊C412yes—
70C2×D4×C23—21no—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23C22⋊C412yes—
74C22×C4×C23—22no—
75C22×C4×C23—21no—
76C22×C4×C23—22no—
77C22×C4×C23—24no—
78C22×C4×C23—41no—
79C22×C4×C23—81no—
80C22×C4×C23—81no—
81C22×C4×C23C2×D412yes—
82C22×C4×C23C2×D411yescharacteristic
83C2×C8×C23—22no—
84C2×C8×C23—24no—
85C2×C8×C23—42no—
86C2×C8×C23—41no—
87C2×C8×C23—42no—
88C2×C8×C23—41no—
89C4⋊C4×C23—28no—
90C4⋊C4×C23—24no—
91C4⋊C4×C23—24no—
92C4⋊C4×C23—24no—
93C4⋊C4×C23—24no—
94C4⋊C4×C23—22no—
95C4⋊C4×C23—22no—
96C4⋊C4×C23—41no—
97C4⋊C4×C23—41no—
98C4⋊C4×C23—44no—
99C4⋊C4×C23C22⋊C414yes—
100C4⋊C4×C23C22⋊C414yes—
101C22⋊C4×C23—28no—
102C22⋊C4×C23—24no—
103C22⋊C4×C23—24no—
104C22⋊C4×C23—24no—
105C42×C23—24no—
106C42×C23C22×C411yescharacteristic
107C42×C23C2×D412yes—
108C42×C23C2×D411yescharacteristic
109#8507C2311yesmaximal, characteristic, direct product factor, Sylow
110C23×C23—22no—
111C23×C23—41no—
112C23×C23—42no—
113C23×C23—41no—
114C23×C23—42no—
115C23×C23#12011yescharacteristic
116Q8×C23—42no—
117Q8×C23—81no—
118Q8×C23—84no—
119D4×C23—42no—
120D4×C23—81no—
121C2×C4×C23—21no—
122C2×C4×C23—22no—
123C2×C4×C23—21no—
124C2×C4×C23—22no—
125C2×C4×C23—24no—
126C2×C4×C23—22no—
127C2×C4×C23—24no—
128C2×C4×C23—42no—
129C2×C4×C23—41no—
130C2×C4×C23—42no—
131C2×C4×C23—42no—
132C2×C4×C23—41no—
133C2×C4×C23—42no—
134C2×C4×C23—84no—
135C2×C4×C23—81no—
136C2×C4×C23—82no—
137C2×C4×C23C2×C22⋊C412yes—
138C2×C4×C23C2×C22⋊C411yescharacteristic
139C2×C4×C23#12012yes—
140C2×C4×C23#12011yescharacteristic
141C8×C23—42no—
142C8×C23—44no—
143C2×#563C2×C2311yescharacteristic
144#1992C2×C2312yescomplemented
145#1959C2×C2311yescharacteristic
146C2×#330C2×C2312yescomplemented
147C2×#329C2×C2311yescharacteristic
148C22×C23—42no—
149C22×C23—44no—
150C22×C23—81no—
151C22×C23—81no—
152C22×C23—82no—
153C22×C23#37911yescharacteristic
154C22×C23#45311yescharacteristic
155C22×C23#45412yes—
156C22×C23#45511yescharacteristic
157C22×C23#45612yes—
158C4×C23—21no—
159C4×C23—21no—
160C4×C23—22no—
161C4×C23—22no—
162C4×C23—44no—
163C4×C23—42no—
164C4×C23—44no—
165C4×C23—161no—
166#563—24no—
167#563—22no—
168#563C4×C2314yescomplemented
169C2×C4⋊Q8—21no—
170C2×#126—22no—
171C2×#125C22×C2312yes—
172C2×#124C22×C2311yescharacteristic
173C2×#123—21no—
174C2×C22⋊Q8—22no—
175C2×#103—21no—
176C2×#102—21no—
177C2×#97C22×C2311yescharacteristic
178C2×#97C22×C2312yes—
179#384C22×C2311yescharacteristic
180#340—22no—
181#340—22no—
182#330—24no—
183#329—22no—
184C2×C23—82no—
185C2×C23#165711yescharacteristic
186C2×C23#166011yescharacteristic
187C2×C23#166412yes—
188C2×C23#179211yescharacteristic
189C2×C23#179312yes—
190C22×Q8—21no—
191C22×Q8D4×C2311yescharacteristic
192C22×D4D4×C2311yescharacteristic
193C23×C4—21no—
194C22×C8—22no—
195C22×C8—21no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198#126—28no—
199#126—28no—
200#125—22no—
201#125—22no—
202#125C2×C4×C2314yes—
203#125D4×C2314yes—
204#124—22no—
205#124C2×C4×C2312yes—
206#124D4×C2312yes—
207#123—24no—
208#123—24no—
209C22⋊Q8—28no—
210C22⋊Q8—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4D4×C2312yes—
217C2×C4⋊C4D4×C2312yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C42—21no—
222C2×C42C23×C2311yescharacteristic
223#103—42no—
224#102—42no—
225#97—22no—
226#97—22no—
227#97—22no—
228C23#850711yescharacteristic, direct product factor, Sylow
229C24—21no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—81no—
235C2×Q8—81no—
236C2×Q8C22⋊C4×C2312yes—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C22⋊C4×C2312yes—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—41no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4C2×D4×C2312yes—
249C22×C4C2×D4×C2311yescharacteristic
250C2×C8—22no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—41no—
254C2×C8—42no—
255C2×C8—41no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—44no—
266C4⋊C4C22⋊C4×C2314yes—
267C4⋊C4C22⋊C4×C2314yes—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C42—24no—
273C42C22×C4×C2311yescharacteristic
274C42C2×D4×C2312yes—
275C42C2×D4×C2311yescharacteristic
276C23—22no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23C23×#12011yescharacteristic
282Q8—42no—
283Q8—81no—
284Q8—84no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—84no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4C2×C22⋊C4×C2312yes—
304C2×C4C2×C22⋊C4×C2311yescharacteristic
305C2×C4C23×#12012yes—
306C2×C4C23×#12011yescharacteristic
307C8—42no—
308C8—44no—
309C22—42no—
310C22—44no—
311C22—81no—
312C22—81no—
313C22—82no—
314C22C23×#37911yescharacteristic
315C22C23×#45311yescharacteristic
316C22C23×#45412yes—
317C22C23×#45511yescharacteristic
318C22C23×#45612yes—
319C4—21no—
320C4—21no—
321C4—22no—
322C4—22no—
323C4—44no—
324C4—42no—
325C4—44no—
326C4—161no—
327C2—82no—
328C2C23×#165711yescharacteristic
329C2C23×#166011yescharacteristic
330C2C23×#166412yes—
331C2C23×#179211yescharacteristic
332C2C23×#179312yes—
subgroupG/H
center, socleC23×C23#120
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C42×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×#1992C2semidirect
4. C2×C23×#330C2semidirect
8. C23×#563C4semidirect
109. #8507C23direct (N × H)
144. #1992C2×C23semidirect
146. C2×#330C2×C23semidirect
168. #563C4×C23semidirect
228. C23#8507direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C23×#5632
C23×#19922
C23×#19592
C2×C23×#3302
C2×C23×#3292
#850723